Collaborative Research: Mechanics of Knots and Tangles of Elastic Rods

在从航运到航海再到手术的环境中，每天都要使用数千种不同类型的结，每种结都需要特定的力量来收紧。该力取决于材料属性、摩擦力和结的拓扑结构。线结通常还具有承载能力;超过该水平的力会导致线结松开，并且过度的力可能导致线结中的材料失效。此外，有些结在没有任何外力的情况下紧紧地抓住，而另一些结则很容易解开。换句话说，这些结可以在材料中储存能量。这个概念被称为拓扑电池，其含义包括DNA中的纳米尺寸结到结构工程中的宏观结。该奖项支持研究，以了解结的基础科学。这项工作将开发模型和计算方法，用于分析结和缠结的力学。同时，它将制定实验技术，系统地研究这种机制。该研究将通过开发教学工具（视频，笔记和演示）为本科生和研究生课程的补充。计算软件也将公开提供。本计画之研究目的为量化弹性杆系结之力学反应。该项目将采用（1）受计算机图形学启发的快速数值模拟，（2）具有可定制摩擦力的创新材料，以及（3）自主机器人实验来解开结的力学。即使是最基本类型的结（上手结），打结所需的力也取决于（1）弹性、（2）摩擦和（3）拓扑结构的复杂相互作用。有趣的是，上手结可能会经历一个通过屈曲不稳定超过一个临界量的拉力。基本纽结的这种不稳定性表明了纽结的力学行为的丰富性。在开发仿真和实验工具后，将研究几种常见结（例如上手结和鞋带结）的机械响应和不稳定性。利用仿真工具的计算速度和机器人实验的自主性，将量化几种类型的结的机械响应，以建立它们的力学库。这些数据将被用来合理化的拓扑，材料和摩擦参数的函数结的机械响应的变化。类似于元素周期表，将制定一个基于力学的节分类方案，其中节将分为各种类别，如，摩擦主导的节，弯曲主导的节，和其他。该奖项反映了NSF的法定使命，并已被认为是值得支持的评估使用基金会的智力价值和更广泛的影响审查标准。

In settings from shipping to sailing to surgery, thousands of different types of knots are used every day, each requiring a specific amount of force to tighten. This force depends on the material properties, friction, and the topology of the knot. A knot typically also has a load-bearing capacity; force beyond this level causes the knot to be undone, and excessive force may result in material failure in the knot. Moreover, some knots hold tight without any external force while others easily get untangled. In other words, the knots can store energy in the material. This concept is called a topological battery with implications in nanometer-sized knots in DNA to macroscopic knots in structural engineering. This award supports research to understand the fundamental science of knots. The work will develop modeling and computational methods for the analysis of the mechanics of knots and tangles. In parallel, it will formulate experimental techniques to systematically study this mechanics. The research will be complemented by developing teaching tools (videos, notes, and demonstrations) for undergraduate and graduate courses. The computational software will also be made publicly available. The research objective of this project is to quantify the mechanical response of knots tied in elastic rods. The project will employ (1) fast numerical simulations inspired by computer graphics, (2) innovative materials with customizable friction, and (3) autonomous robotic experiments to untangle the mechanics of knots. Even in the case of the most basic type of knots (overhand knots), the force required to tie the knot depends on an intricate interplay of (1) elasticity, (2) friction, and (3) topology. Interestingly, the overhand knot may undergo a snap-through buckling instability beyond a critical amount of pull. Such instability in a basic knot points to the richness of the mechanical behavior of knots. After developing simulation and experimental tools, the mechanical response and instabilities of a few common knots, e.g. overhand and shoelace knots, will be investigated. Exploiting the computational speed of the simulation tool and autonomy of robotic experiments, the mechanical response of several types of knots will be quantified to build a library of their mechanics. This data will be used to rationalize the variation of a knot’s mechanical response as a function of the topological, material, and frictional parameters. Similar to the periodic table of elements, a mechanics-based classification scheme of knots will be formulated, where the knots will be grouped into various classes, such as, friction-dominated knots, bending-dominated knots, and others.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.